In this paper we develop a methodology for conducting inference on extreme quantiles of UIH on the basis of noisy estimates, and we establish sharp conditions under which noisy estimates of UIH are informative about the quantiles of interest.
The noisy estimates can be used for estimation and inference if the tails of their distribution converge to the tails of the distribution of interest in a certain weak pointwise sense.
We construct confidence intervals for extreme quantiles of UIH using self-normalizing ratios of extreme or intermediate order statistics of the estimates. We also provide hypothesis tests about the support of UIH.
To obtain the relevant asymptotic distributions, we prove appropriate extreme and intermediate value theorems (EVT and IVT) for noisy estimates.For an introduction to extreme value theory see, among others, [1]}, [2]}, [3]}. See chapter 2 of [3]} for an introduction to the limit theory of intermediate order statistics (also called intermediate extreme order statistics).
